Homotopy coherent category theory and A -structures in ...dodo.pdmi.ras.ru/~topology/books/batanin.pdf · categorical algebra, whereas the alternative ones use the methods of homotopy

JOURNAL OF PURE AND APPLIED ALGEBRA

ELSEVIER Journal of Pure and Applied Algebra 123 (1998) 67 103

Homotopy coherent category theory and A -structures in monoidal categories

Mikhail A. Batanin* Macquaire University, School of' Mathematics and Physics North Ryde, NS W 2113, ,4 ustralia

Communicated by J.D. Stasheff; received 21 March 1995; revised 10 January 1996

Abstract

We consider the theory of operads and their algebras in enriched category theory. We introduce the notion of simplicial A~c-graph and show that some important constructions of homotopy coherent category theory lead by a natural way to the use of such objects as the appropriate homotopy coherent counterparts of the categories. @ 1998 Elsevier Science B.V.

1991 Math. Subj. Class.." 18D20, 18D35, 18{330

0. Introduction

Let A be a small simplicial category, and let F, G : A ~ K be two simplicial functors

to a simplicial category K. Then we can consider, as the simplicial set o f coherent

natural transformations from F to G, the coherent end [8, 10, 12] (see Definition 6.2):

Coh(F, G) = .fA K(F() : ) , G(2)) .

It is well known that in general we cannot compose this transformation. A composit ion

is defined only under some additional conditions [2, 10, 12] and is associative only

up to homotopy. So i f we want to obtain a category structure on the set of coherent

transformations we have to pass to the homotopy classes o f these transformations. The

resulting category is no longer simplicial, or more precisely has a trivial simplicial

structure.

* E-mail: [email protected].

0022-4049/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved PH S0022-4049(96)00084-9

68 M.A. Batanin/Journal of Pure and Applied Alyebra 123 (1998) 6~103

This phenomenon is of very general nature. It is sufficient to note that in the theory

of simplicial closed categories of Quillen [22] the passage to the homotopy category Ho(C) leads us to the losing of the initial simplicial structure.

Our attention to this phenomenon was motivated by the problem formulated by Bourn, Cordier and Porter, concerning categorical interpretation of strong shape theory [6, 11, 12]. Roughly speaking, the problem is to define the strong shape category for any simplicial functor and to compare such categories under the appropriate conditions.

Here the situation is analogous to that just described. The primary source of diffi- culties lies in the fact that we cannot construct any reasonable comparison functor between strong shape categories, because the passage to the homotopy classes leads to the losing of initial enriched structure. So we have to find some structure on the set of strong shape morphisms, which could play the role of simplicial structure on the category. Of course, we have to generalize also the notion of simplicial functor and to demand that this "functor" is isomorphism provided it is bijective on the objects and induces the homotopy equivalences of the "spaces of morphisms".

There is an immediate analogy between the problems above and the problems arising in the loop spaces theory and delooping machines [5, 20, 25, 27]. The purpose of this paper is to reveal this analogy and to provide some technical tools for constructing and working with this generalized category theory. As a line of attack we choose the theory of A~-operads and their algebras developed by May [20].

Our main idea is to replace the notions of simplicial category and simplicial functor by the notions of simplicial A~-graph and coherent morphism between A~-graphs. In some sense our paper extends the article [12], which considers the phenomenon of homotopy coherence on the level of coherent diagrams and their natural transformations. It is our opinion that if we want to obtain a closed variant of homotopy coherent category theory, we have to replace the categories by their homotopy coherent counterparts. In support of this statement we prove that many of the "homotopy coherent categories" are homotopy categories of some locally Kan A~-graphs. In particular, the "H-category" structure on the set of coherent transformations introduced in [10, 12] may be extended to an A~-structure (Theorem 6.3), at least for the case when K is a

locally Kan category. It should be pointed out that our approach is not the only possible. Another idea,

which goes back to Boardman and Vogt [5, 29], is to use weak Kan complexes as the appropriate generalization of the notion of category. The reason is that in such a complex one can define the "compositions" of the chains of the 1-simplices, which is associative up to all higher homotopies. This notion lies at the heart of the Cordier-Porter methods. Recently it was used successfully by Gfinther [15] in strong shape theory.

We should mention also a paper [23] of Schw~inzl and Vogt, where the ideas related

to Segal' delooping machine were used. Each of these methods has its own advantages and disadvantages, which we do not

discuss here. Note only that our approach allows to call on the powerful methods of categorical algebra, whereas the alternative ones use the methods of homotopy theory.

M.A. Batanin/Journal of Pure and Applied Algebra 123 (1998) 67-103 69

We do not consider here the applications of our theory to the strong shape theory. It is the theme of a separate article [3].

Some words about the notations and terminology. If .,~¢ is a monoidal category and K is a category enriched in J then K will mean the ,~-enriched hom-functor in K [16].

An additional point to emphasize is that we must take some extra care concerning the size (in the set theoretical sense) of the objects under consideration. In our paper we are taking the view of Kelly on the existence of ~/-categories of ~'-functors (see the discussion on pp. 69-70 in [16]).

We now will undertake a brief overview of the paper. Section 1 is devoted to the basic notions and definitions. We have to generalize

May's initial approach in such a way that it becomes applicable in some enriched monoidal categories. Although it is sufficient for our applications to consider the topological and simplicial categories, we prefer to work in maximal generality having in view further applications to the categories enriched in the other monoidal categories (such as categories of groupoids, of categories and various categories of algebraic

nature). In our paper we concentrate attention on the monoidal categories, which are not

symmetric. So the permutations may be disregarded in the theory. Thus our notion of operad is an immediate generalization of May's non Z-operad. However, there is some difference, because we work without base points. So we do not require the triviality of the 0-term of our operads. This difference affects also the construction of a monad

corresponding to an operad. It differs from that of May and agrees with the monad C+

introduced by Yhomason in [27]. In Section 2 we construct a homotopy theory of algebras over an operad and consider

the problem of "rectification" with respect to A~-monoid structure. The main results here are Theorems 2.3, 2.4 and Corollary 2.3.1 giving us the conditions under which an A~-monoid is isomorphic in some homotopy coherent sense to a monoid.

In Sections 3 we develop some technique of lifting of algebra structures with respect to a map of operads. We prove here that there is some universal A~:-operad, which acts on every A~-monoid. In Section 4 we prove a variant of the theorem of homotopy invariance of the Ao~-monoid structure. The variants of the results of Sections 3 and 4

are well known in the theory of loop spaces [5, 17, 25]. In Section 5 we establish two main technical results, which give us a powerful

method for constructing the A~-monoids in simplicial and topological monoidal categories. The first of these results (Theorem 5.1) asserts that the standard cosimplicial topological space A has a structure of an A~-comonoid. This Ao~-comonoid structure is a generalization of the rule of composition of higher homotopies considered in [2, 9, 18]. The second (Theorem 5.2) shows that the cosimplicial realization of a cosimplicial A~-monoid with respect to a cosimplicial A~-comonoid is again an A:~-monoid.

In Section 6 we consider some applications of the results obtained to the homotopy

coherent category theory. Our main goal here was to demonstrate that the A~-structures arise in a natural way in many important fields of category theory such as bar and

70 M.A. Batanin/Journal of Pure and Applied Algebra 123 (1998) 67 103

cobar constructions and the above-mentioned problem of composition of coherent trans-

formations.

1. Operads and their algebras in enriched monoidal categories

Let ~/ be a closed symmetric monoidal category with tensor product ®,~, unit object I and with initial object 13 and let o~/ be complete and cocomplete [16]. We denote by ~ the internal hom-functor on ~r;/. Let ,A" be the set o f natural numbers.

Definition 1.1. We call the category of ~/-collections the ~/-category o f ,~/-functors

from (with discrete category structure) to ,~. We shall denote it by Coll.

So the objects of Coil are the sequences of the objects of ~ ' , and the enriched

hom-functor from A = (Ao, A1 . . . . ) to B = (Bo,B1 . . . . ) is

oc

Coll(A, B) = ~I.~/(Ai,Bi) . i=0

Definition 1.2. Let K be an ~;/-category with a bifunctor ~K : K ®.~j K ---+ K, an object

IK of K and isomorphisms

aA,&C : (A ~'K B) ~ - C --+ A ~X (B ~)~ C),

IA : IK @K A ---~ A,rA : A @K [K ----~ A,

which make K a monoidal category [16]. We say that K is a monoidal ~;/-category if

~K is an ,~'-functor and a, l, r are sO-natural.

For example, the category Coil has the two following monoidal structures:

1. The fibrewise monoidal structure with fibrewise tensor product, which we shall

denote , ~ / , as well, and with ,,~/= (1,I . . . . ) as a unit object.

2. For two collections A and B put

(A '~'Col! B)j = H H A j, (~i~¢ . . . . ~.~¢ Ajk @=¢ Bk k > l jL + ' " + J ~ - J

for j > 1, and

(A :gColl B)o = / ~ 0 I I I I , A 0 , ~ / ,,~,, Ao ~.~¢ Bk. k>>l

Let Ico# be a collection (13,L13 . . . . ).

Remark. A similar tensor product of the collections was introduced by Smirnov in [24]. The basic difference is that our notions o f collection and tensor product involve the

0-dimension terms.

M.A. Batanin/Journal of Pure and Applied Algebra 123 (1998) 67 103 71

Proposition 1.1. The categot 3, Coil with G'cotl and Icol/ above is closed on the right monoidal .~J-eategorv with internal hom-functor

Coil • Coll °p ~.~./ Coil ~ Coil,

Coll A,B)k : H I-[ . . . .

j>O jt +'"+jl,=j

(That is

Coll(A ~>CoZt B, C) ~- Coll(B, Coil(A, C)), VA, B, C E Ob(Coll).)

In addition, there is an ~-natural transJbrmation

(A ~,.~/ B),~,co:/(C ~ / D) ---+ (A~,co]:C)~.~/ (B'~:couD)

and a morphism fool /~ ...//, which make ~.c/ a monoidal functor [13, 5].

Proof. The proof consists in immediate verification. []

Further we understand the category Coll as a monoidal category with respect to the

second structure and denote '~'coH and Ico/: by ~> and 1 respectively if it leads to no

confusion.

Definition 1.3. We call an ~'-operad ~ with multiplication

and unit

I+7. I--> E

a monoid (E,~,,t/) [19] in Coll.

We shall denote by Oper the ~/-category o f operads.

Examples. 1. For any ,.~¢ the collection o./ /= (I, I . . . . ) has a natural ~]-operad structure. 2. Let K be a monoidal s~/-category with enriched hom-functor K, tensor product

~ x and unit object Ix. For X E Ob(K) put

X j = x ~ > x . . . ~ x X , j ¢ O , X ° = I x .

I

Then one can associate with X two ~-operads End(X), End°(X):

End(Y) j --- K_(xJ,x), End°(Y)j -~ K(X ,X j )

called the operad of endomorphisms of X and dual operad o f endomorphisms of X.

72 M.A. Batanin/Journal of Pure and Applied Algebra 123 (1998) 67 103

For some technical purposes we need the following generalization of the example 2 above. In the situation of the example consider two .s~-functors:

~ , ~_o : K °p ,+~.<" :1 A" --+ Co l l,

~_(x, Y) j = K_(X:, Y), ~_°(X, r ) : = K(X, y l ) .

There are also two obvious natural morphisms:

: ~ ( x , Y ) ~ ~_( Y, z ) ~ ~_(x, z ) ,

~o . -K"(x, Y) ,~ ~ ° ( Y , z ) ~ ~_° ( x , z ) .

In addition, for each object X of K we have two morphisms in Colh

• / -~ ~_(x,x), ~o. / _~ ~_O(x,x )

generated by an ~ -enr i ched structure on K.

Proposit ion 1.2. The functors ~ (H--°), natural transJbrmation t~ (I ~°) and morphism

e (~o), define some enrichment o f K in Coll.

Proof. Immediate from the definitions. []

Let g be an ~J-operad.

Definition 1.4. An g-algebra in K is a pair (X, q), where X is an object o f K and ~/

is a morphism of .~-operads

~I " g ~ End(X) .

So our definition of an algebra of operad is the same as in [20].

Definition 1.5 (Smirnov [24]). An g-coalgebra in K is a pair (X,p) , where X is an

object of K and p is a morphism of ~ - o p e r a d s

p • g ---+ End°(X) .

Usually, we write X for an g-algebra (g-coalgebra) (X, ~/) assuming that we know the structure map r/. We write also Alg: and Coalg:' for the ~/-categories of g-algebras

and g-coalgebras correspondingly.

Example . For any monoidal °~/-category K the category of ..///-algebras is isomorphic

to the category of monoids in K. Let K be a monoidal .~-category and let K be .~-tensored (s¢-cotensored) by

( - ) ®.uz ( - ) : K,~, ~: ~/ ---+ K ( ( - ) ( - ) " sg°P®.~K ---+ K).

M.A. Batanin/Journal of Pure and Applied Algebra 123 (1998) 6~103 73

In general, there is no connection between this functor and tensor product 'g,c in K. But it will be convenient for us to assume that ( - ) ~ , ~ ( - ) ( ( - ) ( ~ ) is monoidal. That is for such a tensorization (cotensorization) we have an ,~-natural transformation

7 : (X~55.~./A) ~ x (Y ®.~¢B) ~ (X ~ x Y) ~.~J (A ~ / B ) ,

(o) : X A '~K yB -+ (X ®K y)a @,~/ B)

and a morphism

7; : Ix --~ IK ~.~/ I (Oal : IK ---+ I](),

which satisfy the axioms of monoidality [13, 5].

We shall say also that K is strongly ,~-tensored (s / -cotensored) provided 7 and 7I (~,) and col) are isomorphisms.

Proposit ion 1.3. Let K be a monoidal ~/-cate,qory with coproducts, eommuting with tensor product. I l K is ag-tensored, then the formula

[~] (X) = I I ( x " ) , ~ , , ~ , ~ ~ Ob(Coll), X C Ob(K) n>O

defines an ~o/-Jhnctor Ji'om Coil ~ / K to K. Ijl in addition, ~ is an ~eg-operad then the funetor

[ ~ ] ( - ) : K -~ K

has a natural structure of the sJ-monad. The cate,qory of al,qebras of this monad is isomorphic to the cate,qory of g-al,qebras. Furthermore, if K is stronqh, ~d-tensored, then [ - ] ( - ) ' Coll ~.~j K --+ K provides the Coll-tensorization of the Coil-enrichment

K o f K .

Proof. The proof is well known [20]. []

Examples . 1. Let K be a monoidal ,~-category with coproducts and zero object 0 (that is G~K13 ~-- (~,~KG ~ 13 for any G E Ob(K)) and let :~x commute with coproducts. The category Gr(C) of K-graphs over a fixed set o f objects C is the following ,~/-category. The objects o f Gr(C) are the functions:

G : C × C --+ Ob(K).

The enriched hom-functor is

Gr(C)(F,G) = H K(F(a,b) ,G(a,b)) . a, bGC

The monoidal structure is given by

(G ~Gr(C') F)(a, b) = H G ( a , C)@KF(C, b) c~C

74 M.A. BataninlJournal o.1' Pure and Applied Algebra 123 (1998) 67 103

and

1G,.(c)(a,b) = 1 if a = b, Ia,(c)(a,b) = 0 if a ¢ b.

It is clear that Gr(C) is strongly .~/-tensored and sJ-cotensored if K is. Note that

the category of monoids in the category of ~ - g r a p h s is isomorphic to the category of ~/-categories with fixed set of objects C and ~¢ff-functors between them, which are identities on C.

2. If C is an ~ ' -category, then the category of ~.~-endofunctors F(C, C) is monoidal ~</-category with respect to the composition. It is ~/-tensored (~ -co tensored) if C is ~4-tensored (~/-cotensored). The category of monoids in F(C, C) is isomorphic to the category of s//-monads on C.

3. Let C be an .;J-category. Then we can consider the ~ -ca t ego ry Dist(C, C) of sff-distributors (or profunctors) from C to C [4], that is the .4-category of ,v/-functors

G : C°P,~,~/ C---~ ~/.

The tensor product is defined by the coend:

P c

(a ~D~s~(C,C~ F)(X, Y) = J G(X, C)~,/F(C, Y)

and unit object is

Ioist(C,C) = C : C °p @~.:t C --~ .~/.

As in the example 1 Dist(C.C) is strongly ~/-tensored and .¢ff-cotensored. It is clear that the example 1 is a special case of 3 and the functor of discretization

(_ )d induces a monoidal ~/-functor:

Dist(C, C) ~ Dist(C d, C d) ~ Gr(Ob(C)) .

We have also a fully faithful strict monoidal s ] -embedding [4, 6, 11]:

~ , : F(C, C) ~ Dist(C, C), d?L(X, Y) = C(X, LY) , L C Ob(F(C, C)).

So the example 2 is connected closely with 3 as well. There is also another embedding [4, 6, 11]:

~* :F(C,C)°P---+Dist(C,C), O L ( X , Y ) = C ( L X , Y), L E O b ( F ( C , C ) ) ,

with similar properties. This provides us with the following generalization of a Bousfield-Kan lemma [7, p. 27; 2, p. 283].

Proposition 1.4. Let o ~ be an .y/-operad. Then an endojimctor R : C ~ C (L " C --~ C) has an N-algebra (d-coalgebra) structure i f and only i f the distributor OR (d? L) has this structure.

4. More generally, let K be a cocomplete monoidal sff-category with strong ~ - tensorization, and let '~'K commute with colimits. I f C is an d -ca tegory , then we can

M.A Batanin/Journal of Pure and Applied Alyebra 123 (1998) 6~103 75

consider the following category of K-distributors Distx(C, C). The objects are the sC- functors from C °p ®.~j C to K. The morphisms are their ,~-natural transformations. The unit object is defined by

IDist~(<c)(X~ Y) : IK ,~'~ rj f ( X , Y ).

Tensor product is defined as in example 3 with @x instead of @.:j.

Below we assume that the category .~ has a structure of a closed model category in the sense of Quillen [22].

Examples. 1. The cartesian monoidal categories: - of simplicial sets J , - of compactly generated topological spaces J~p [26], - of (small) categories <6at, - of groupoids Gpd.

2. The various categories of algebraic nature (see [14, 24]). In this case the category Coll has an obvious fibrewise closed model category structure.

Definition 1.6. A morphism of ~/-operads is a local weak equivalence, if it is a weak equivalence of the corresponding ~'-collections.

Definition 1.7. An ~J-operad ~ together with a morphism g--~ ~ [ /o f .~'-operads (augmentation), which is a locally weak equivalence, is an A~-operad. A morphism of A~-operads is a morphism of operads commuting with augmentations.

Definition 1.8. We say that an d°-algebra (g-coalgebra) X in K is an A~-monoid (A~:-comonoid) if E is an A~-operad.

In the category of aJ-graphs we give a separate definition.

Definition 1.9. An A:~-graph is an A~c-monoid in Gr(C).

2. Coherent homotopy theory of E-algebras

In this section we assume that the category ag has also a simplicial enrichment -qL/

such that

~ ( x , Y) ~- ~ , ( L + , ~ ( x , Y))

for any X, Y E Ob(xg) and such that it is the monoidal J -ca tegory with strong finite ~-tensorization - x - . Furthermore, the simplicial category ~A/ has a structure of simplicial closed model Quillen category [22]. The categories from the example 1 of

the previous section are of this type.

76 M.A. Batanin/Journal of Pure and Applied Algebra 123 (1998) 67-103

Definition 2.1. An °~'-category K is locally fibrant if K(X, Y) is a fibrant object o f ~4 for all X, Y E Ob(K).

For a simplicial category C we denote by ~(C) its homotopy category, that is ~z(C) has the same objects as C and

~ ( c ) ( x , Y) = ~zo(c_(x, Y)).

For any ~4-category K we can define an "underlying" 5~-category K~/' by

K,f(X, Y) = .~w.(Io/,K(X, Y)).

If, in addition, K is .~/-tensored by - , g . ~ e - , then K~/ is finitely JJ-tensored by

X × k = X , ~ . / ( L j ×k)

for a finite simplicial set k. We shall denote by u(K) the category u(K~').

Lemma 2.1. I f the unit object of ,~ is cofibrant then for any locally fibrant category K the underlying 5~-categot~ ' Kv' is a locally Kan 5~-categoo,.

Recall now a construction from [2]. A generalization of this construction will be considered in Section 6.

Let (L,/~,e.) be an .~-comonad on K. Then for every X E ob(K) one can define a simplicial object L . ( X ) of K putting

Ln(X)=Ln+I(x) , d i = L " i 'g. 'Li , s , = L " i.IL.Li. (1)

Suppose, now, that K,/ is finite 5f-tensored and there exists the realization

f' L ~ ( X ) = L,,(X) x A".

We thus obtain an ~-endofunctor on K(/~ which we shall denote by L~ . The counit e generates an 5f-natural transformation

~;~ : L ~ - + L

By applying the functor K, / , ( - , Y) to L . ( Y ) levelwise we obtain a cosimplicial object

K~,(L,(X), Y) in 5f and

K__~t(L~(X), Y) ~_ Tot(K__.,/;(L,(X), Y)).

where Tat is the Bousfield-Kan total space functor [7]. Let K ~ be a full subcategory o f K. As was proved in [2] (see also Section 6),

if K ~ ( L . ( X ) , Y)) is a fibrant cosimplicial simplicial set in the sense of [7] for every

X, Y E ob(K~), then one can define a category C H L ~ - K ~ called the coherent homotopy category of the comonad (L,/z,~), which has the same objects as K t and

C H L ~ - K'(X, Y) = :~o(Tot(X~(L,(X), Y))) .


If, in addition, there exists L ~ in K' , then one can define a simplicial transformation

I t~ " L ~ ---+ L 2

such that ( L ~ , # ~ , ~:~) becomes a comonad on ~(K~), and C H L ~ - K ~ is isomorphic

to the Kleisli category of this comonad. Let now ~ be an ,.~-operad. Let K be a monoidal ,~/-category with ,~/-tensorization.

Then the monad [g] exists and the category o f E-algebras is isomorphic to the category of algebras of [E]. So we have a pair of ,~/-adjoint functors •

T = [d] " K --~ A lg ~, U "A lg ~ --~ K,

which generates by the usual way the monad on K and the comonad TU

11" T U ~ I, p : TU--- , T U T U

on A l q ~: that we shall denote by [~'] as well. So we have a simplicial object [N] .X in A l g ~" for each g-algebra X, which is none other then the May bar-construction

B, ( [~] , [d ] ,X) [20]. Our first goal is to find the conditions under which Alg~cr([E].X,~. Y ) is fibrant in

the category of cosimplicial simplicial sets c J ~.

For this we need a definition from [21].

Definition 2.2. Let ~ be a closed model Quillen category. Let, in addition, .~ be a monoidal category with respect to tensor product ~.~ and unit object I~. We say that a cofibration f : A ~ X is closed (with respect to ,~2~) if the following condition is

satisfied: For any cofibration

g : B ~ Y

the canonical morphism

(X ,~ ,~B) H (A@.~Y)---+ X , ~ , ~ Y

A@.~4B

is a cofibration.

Note that in J and .Ybp all cofibrations are closed. For 5 r this may be established

immediately, whereas for .Y~p this is one o f the results o f Steenrod [26] (see also [20]). Note also that if a morphism f : A -+ B in C o l l is such that fn is a closed cofibration

for every n _> 0, then it is a closed cofibration in C o l l (with respect to @Coil).

Let now D : Ai - + B i , i = 1,n be a family o f morphisms in .~. Let

Zi = B1 ~ ,x . . . . ~ , ~ B i _ l @.#Ai@'.#Bi+ 1 • " • @ , ~ B n ,

Z O = B l ~ . ~ . . . . ~ ' . ~ A i @ . # . . . . ~,~Ai@.~ • • " (&~Bn, i < j .


Then we can construct the following diagram :

z , . . . z , . . . z ~ . . . &

\ / Z# l<_i<j<n,

where d i = id~:~. . . ,~. ,xf~®.e. . . ®~id. Let D ( f l . . . . , f , ) be the colimit of this dia-

gram. Then the maps

id@~. • • ~.~f-@.~ •. • ~.~id : Zi --+ BI @~ • " • ~,.~B~

induce a canonical morphism

d : D ( f l . . . . . f , ) ---+ BI,~.~. . . @.~Bn.

Lemma 2.2. I f in ~ the tensor product '~'~z commutes on the right with finite eolimits

then Jor any fami ly o f closed cofibrations f,. : Ai --~ Bi, i = 1, n the canonical

morphism

d :D(f l . . . . . f~) -+ Bl~,~ ...,~,sB.

is a eofibration.

Proof . This follows from the inductive arguments since D ( f l . . . . . f i ,) may be obtained

as a colimit of the following diagram:

id@.~d A l ~.~D(f2 . . . . . f,~) > A 1 @,'~B2 ~'.~"" ~,~Bn

BI~'.,3D(f2 . . . . . f~) []

D e f i n i t i o n 2 .3 . An ~¢-operad g is cofibrant provided the unit of

I - - + ~

is a closed cofibration in Coll.

Theorem 2,1. Let K be a cocomplete monoidal ,~/-category with strong ~- tensori -

zation. Let ~ be a cofibrant ,~-operad and let X and Y be two 8-algebras.

Then the cosimplicial simplicial set

A lg~..9,,([ ~].X, Y)

is fibrant, (f ~(X, Y) is a fibrant ,z/-collection.

Proof. Let K,L be two ~4-categories and let K be cocomplete. Let S : K --~ L be

an ,~-functor having the right ~/-adjoint T : L --~ K. Then we have an ,~ -comonad


A = ST on L and an s~/-monad B = TS on K. For X, Y E Ob(L) consider a cosimplicial

object in ~ ,

U* = L_(A.X, Y), U" = L(A,,+~X, Y).

Let M " ( U * ) be the limit o f the diagram

u,7 . u , " .

U, S O<i<j<n .

Then the codegeneracy morphisms s i" U n+j ---, U n induce a morphism

r~ . U,-1 ~ M~(U*) .

Let Z* be an augmented cosimplicial object B*TX, Z ~ = B n+l TX with cofaces and

codegenaracies generated by the monad B on K and let D"(Z*) , n >_ 0 be the colimit

of the diagram

n n Z,I Z" Z ~ "'" i "'" j "'" Z,-I

"~ 0<i<j_<n+l.

For n = - 1 we put o - l ( z *) = Z -1. Then the coface morphisms d i ' Z," -~ Z ~+1

induce a canonical morphism

pn :Dn(Z *) --+ Zn+l.

We have the following lemma:

Lemma 2.3. The canonical morphism

r n • U n + 1 ____+ M n ( U *)

is the result o f application o f the func tor K ( - , TY) to the morphism

p~- i . D ~ - I ( Z *) ~ Z".

Proof . The statement is evident in view of the isomorphisms,

U n = L(An+IX, Y) ~ K(BnTX, TY) = K ( Z ~-1, TY),

and the fact that codegeneracies in U* are induced by cofaces in Z*. []


Apply now this lemma to the pair of ~4-adjoints

T = [ 8 ] : K - - * A I 9 ~, U : Alg ~ ~ K.

So the morphism

r n : M"(Alg~([~] ,X , Y ) ) -+ Alg~'([,~]n+Zx, Y)

is the result of application of K ( - , UY) to

p,~-I : D,~-~([E]*UX) ~ [o~],+1 UX.

But

K([oz]k+lX, Y) ~- Co l l (~ k+j, ~(X, Y)),

K(D"-I([~]*X), Y) ~_ C o l l ( D " - I ( ~ * ) , ~ ( X , Y ) )

and so r" is the result of the application of C o l I ( - , K ( X , Y ) ) to

6" I :D ~-jS* ~ + ' .

The last is a cofibration in Coll by Lemma 2.2. Now Theorem 2.1 follows from the fibrantness of K(X, Y). []

Let K be a cocomplete monoidal ,4-category with strong ~/-tensorization and let K ~ be its full locally fibrant subcategory. Let ~ be a cofibrant ,e/-operad. Theorem 2.1 allows us to give the following definition:

Definition 2.4. We call the coherent homotopy category of g-algebras in K ~, CH Alg e' (K') , the coherent homotopy category

CH [8] ~ - A lg ~ (K') ,

where AlgZ'(K ') is the full subcategory of Alg ~ generated by objects of K ~. If K itself is locally fibrant, then we write simply C H A l g ~" for C H A l g ~ ( K ) .

Let now iV, be a simplicial object in K. If K is cocomplete, then we can consider the realization functor I - ] :

j k

Ix , I = x~ × A k.

There exists an obvious natural morphism

Ix , ~ , Y,I ~ Ix, l~KIr, I,

where @, is the fibrewise tensor product of simplicial objects. We shall say that ®x commutes with realization if it is an isomorphism.

M.A. Batanin/Journal of Pure and Applied Algebra 123 (1998) 6~I03 81

Theorem 2.2. The canonical Junctor

P : ~Alge~ ---~ CH AIg ~

inverts the morphisrns of ?-algebras, which are the homotopy equivalences in K. Furthermore, P is the localization fimctor w'ith respect to the above class of the

morphisms of ?-algebras, provided Ks, is cocomplete and @K commutes with realization.

Proof. The proof follows from Theorem 2.1 and the results of [2]. Indeed, if ®K commutes with realization, then for any simplicial object X. in K there is an 5 ,~- natural isomorphism

I[?Y~* I ~- [?] Ix*l.

Thus, the comonad ([?]oc,P~, r/~) is defined and we have

-- [ e ] l [ e ] , x l - I [? ] ( [e ] ,X) l .

Hence, the arguments dual to that of [2, Theorem 5.2] give us the desired result. []

Remark. For an ?-algebra X, an ?-algebra [ ? ] ~ X is none other than bar-construction B([?], [? ] ,X) [20].

Corollary 2.2.1. I f @K commutes with realization, then the canonical functor P has a left adjoint Q, which is fully faithful and on the object X is equal to B([g],[E],X). The category CH Alg ~ is the Kleisli category of the comonad on 7zAlg e' generated by this adjunction.

Let now f : g0 --+ ? l be a morphism between two cofibrant d-operads. Then it induces an obvious functor

f * : Alg ~'~ ~ Alg e~°.

On the other hand, if K is coeomplete and ®K commutes with realization, then for every go-algebra X one can consider the bar-construction B([8l], [?0],X), thus obtaining a

functor

f , : Alg ~° -+ Alg e',.

Proposition 2.1. The functors f * and f . induce the functors

F* : CH Alg e~t ~ CH AIg e°, F , : CH Alg e'° ~ CH Alg ~ ,

such that F* is right adjoint to F. . If, in addition, f is a locally weak equivalence, then F . , F * is a pair of inverse

equivalences of categories.

82 ,~I.A. Batanin/Journal of Pure and Applied Algebra 123 (1998) 67-103

Proof. This is an easy consequence of Corollary 2.2.1 and the well-known properties of bar-construction (see the proof of Proposition 1.17 in [27]). The only thing we have to demonstrate is that the cosimplicial simplicial sets K c/ . (B.([gl] ,[Wo],X),Z)

and K,p(B.([E0], [~0],X),Z) are fibrant for every Z E Ob(K). Indeed,

_K(B.([~], [~0],X),Z) -~ K([~j][~0]"X,Z) ~_ Coll([E~][~o]",~_(X,Z)).

Thus the arguments used in the proof of Theorem 2.1 give the desired result. []

Applying this proposition to the augmentation of a cofibrant A~-operad B, we see that the coherent homotopy category of g-algebras is canonically equivalent to the coherent homotopy category of ,//-algebras. Hence, for two cofibrant A~-operads E0, ,~l, there is a canonical equivalence of categories

F : C H A l g ~'~ ~ C H A l g ~;~ .

So one can define for an g0-algebra X and an .~l-algebra Y the set of their coherent morphisms as

CH A lg e~' (F (X) , Y).

It is evident that we thus obtain a category CHMon , with the algebras over different cofibrant A~-operads as objects, which we call the coherent homotopy category of A~-monoids. We summarize:

Theorem 2.3. I f K~ is" cocomplete, locally fibrant and ~'K commutes with realization, then the inclusion

CH AIg / / C CH Mon

is an equivalence o f categories. In particular, every A ~ - m o n o i d is isomorphic in CH Mon to a honest monoid.

Remark. There exists an obvious forgetful functor

U : C H M o n ~ ~t(K),

such that the inclusion C H A l g t e C C H M o n is a functor over ~z(K). However, an inverse equivalence does not. In the following section we construct another A~-operad, for which the corresponding equivalences commute with U.

Corollary 2.3.1. Let all the conditions o f Theorem 2.3 be fulfilled f o r ~¢ itself. Then

the category o f ,~/-graphs over a f i x ed set o f objects C satisfies these conditions too. Hence each Aoc-graph is isomorphic to some ~]-category.

This corollary shows that in many important cases an A~-graph admits a "rectification". It is true, for example, for the category of 3-bp-graphs. Unfortunately, we cannot

M.A. Batanin / Journal of Pure and Applied Algebra 123 (1998) 67-103 83

apply it to the important category of 5P-graphs. However, in this case we can slightly change the proof according to one idea of [12].

Namely, we shall say that an object X of K is fibrant provided the object K__(A,X) of .~/ is fibrant for every A E Ob(K). Let Kf denote the full subcategory of fibrant objects in K. Let, in addition, K be cocomplete and ®x commute with realization; then for every B-algebra X in Kf we can consider the bar-construction B(g ,E ,X) , which is an g-algebra in K. Then the coherent CHAlg~(Kf) is defined correctly and every coherent morphism from X to Y may be specified up to homotopy as the morphism of g-algebras in K:

8(g , ~ , x ) ~ y.

The unit of the monad [C] gives us a morphism ~/:X ~ B ( g ' , g , x ) in K.

Lemma 2.4. Let X and Y be t w o d%al.qebras in K/. And let a morphism of g-al~Tebras O : B ( g , g , x ) -~ Y be such that O.rl is a homotopy equivalence. Then ~ defines an isomorphism in the coherent homotopy category of dg-alqebras in Kf.

Proof. This is evident via Theorem 2.1 and the fact that t I induces a deformation retraction

K ( x , z ) ~- K ( ~ ( ~ , g , X ) , Z )

for every fibrant object Z in K [20, 2, Lemma 2.1].

Return to the case of the category of 5P-graphs. We shall say that J -g r aph G is locally Kan if G(a,b) is Kan for every a,b E C. It is obvious, that the category of locally Kan graphs is the subcategory of fibrant objects in the category of simplicial graphs. If f : g0 --+ BI is a morphism between two .7~-operads, then for every g0- algebra X in Gr(C) one can consider a locally Kan graph

SIB([o~I], [B0],X) I _~ S(B([B1], [B0], [XI)),

where S is the fibrewise singular complex functor and I - I is the functor of fibrewise geometric realization. But S(B([BI], [B0], IX I)) is, obviously, an g l-algebra. Moreover, if f is a locally weak equivalence and X is locally Kan, then the conditions of Lemma 2.4

are satisfed for the morphism

B([~o], [g0],Y) ~ B([g~], [g0],X) ~ S(IB([gl], [g0l,X)l).

Therefore, the conclusions of Proposition 2.1 and Theorem 2.3 remain valid. In partic-

ular we obtain:

Theorem 2.4. Ever), simplicial locally Kan A~-graph is isomorphic in CHMon to a locally Kan simplicial category.

84 M.A. Batanin / Journal of Pure and Applied Algebra 123 (1998) 67-103

3. Lifting of Am-structures

We shall use below some technique which goes back to [5] and is described in a

form convenient for us in [14]. But some modification is necessary, because we want

to work without permutations, but with homotopy units.

A tree is a nonempty finite connected planar (a 1-dimensional subspace of a plane)

oriented graph T without loops, such that there is at least one incoming edge and

exactly one outgoing edge at each vertex of T. Let V(T) ,E(T) be the sets of vertices

and of edges o f T. We say that an edge e o f T is external i f it is bounded by a vertex

at one end only and internal otherwise. The degenerate tree is a tree with a single edge

and without vertices. The notions of output and input edges o f a tree and of a vertex

are the same as in [14]. Every tree has a single output edge, which is bounded by a

vertex called the endpoint of the tree. We denote by In(T), Out(T) the sets o f input

and output edges of a tree 7" and by In(v), Out(v) the sets of input and output edges of a vertex vE V(T).

We fix some orientation on the plane and assume that the orientation of the edges

of a tree T agrees with the direction of the x-axis. So we can consider an order

on In(T) (and on In(v) for every v E V(T)) according to the direction of the y- axis.

We denote by [n] the set {1,2 . . . . . n} with the natural order. A tree T equipped

with an order preserving injection [n] --~ In(T) will be referred to as an n-labelled

tree or simply an n-tree. A 0-tree is a tree without labelling. A degenerate 0-tree will

be denoted by D. A unit tree D I is a l - label led degenerate tree. A tree with a single

vertex and n input edges is called an n-star. Two n-tree T, T ~ are called isomorphic

i f there exists an isomorphism of trees T --~ T / preserving orientations, the order on

In(v),v E V(T) and labellings. Below we identify a labelled tree with its classes o f

isomorphism.

We say that e is a zero edge if it is an input edge without labelling. For any vertex

vE V(T) we denote by In~(v) the set of input nonzero edges at v.

As in [14] the composition 7"1 oi 7"2 of labelled trees is defined. We consider also

another type of composition. Let T be an n-tree and let (T~, v E V(T) ) be a family of

trees, such that T~. is an In'(v)-tree. Then one can define a new tree

T o~cv(r ) T,,

called the composit ion of T and (T,,, v E V(T)). By definition

V(T o~v~r) T~,) = L_J V(T~,). vCV(T)

If vE V(T) is such that T,, is the unit or degenerate tree, then put Z(v) to be the set

of all input zero edges at v. For other ~"s put Z(v) = (0. Then define


where ~ is an equivalence generated by the following relations:

1. O u t ( v ) ~ Out( T~, ).

2. The kth edge in In~(T,,) is equivalent to the kth edge in Int(v). We put, in addition, that a zero edge in In(v) is bounded in Tov~v(r)T,, by the end

point o f T,,.

Let T be an n-tree and e be an edge o f T. I f e is an internal edge or an input

zero edge bounded by a vertex v with In(v) ¢ e then we can form a new tree T/e by

contracting e into a point (see [14]). We assume also that i f a 0-tree T is isomorphic

to a 1-star and e is a single input edge o f T, then T/'e = D. Analogously, i f a 1-tree T

is isomorphic to a l - label led 1-star and e is its single input edge, then T/e = D L. In

all the above cases we say that T/e is obtained from T by edge contraction.

Write T _> T I if T t is isomorphic to a tree obtained from T by a sequence o f edge

contractions. Thus _> is a partial order on the set of isomorphism classes of n-trees.

Let now ~ be an ~ -co l l ec t i on and let S be a finite set with [S[ = k. Then we put

Es = ~ . I f now T is an n-tree we define

~7(T) = ( ~ ) ~ln,(t~) (2) vE V(T)

(compare with formula (1.2.2) from [14]). We assume in addition, that .g(D) =

~ ( D l ) = I . This allows us to introduce the following ,~-functor:

F " Coll ~ Oper,

F(E),, = I_[ g(T), TCTr(n)

where Tr(n) is the set o f isomorphism classes o f n-trees. A multiplication in F ( g )

is induced by the composition oi of trees [14] and a unit is defined by the canonical

inclusion of I as a summand in F ( ~ ) l with unit tree as index. We have also a natural

inclusion o f ,~'-collections

• ~ - ~ F(~),

where qn is the canonical inclusion o f the summand indexed by n-labelled n-star (for

n = 0 this index is 1-star without labelling).

I f ~ is an operad and T > T', then one can define correctly a morphism

as in [14], except for the case when T' = D or T' = D 1. This allows us to define a

natural morphism of operads

; , ' F( ,~) --+ ~.

It is easy to see that F is a left adjoint to the forgetful functor

U " Oper --~ Coll


with q as unit and ~, as counit of the adjunction. This generates an ac/-comonad (F, p, 7) on Oper.

Proposition 3.1. Let ?# and ~' be two operadr, such that ~ is cofibrant and ~ is' fibrant as ~#/-collections. Then the cosimplicial simplicial set

Ope(sF(F*(d$),~ )

is fibrant.

Proof. Let us give the following inductive definition. A 0-stage n-tree is an n-tree.

A k-stage n-tree (or simply (k, n)-tree) (7, ir~, v E V (T) ) is a set consisting o f an n-tree

ir and one ( k - l,In'(t,))-tree T~ for every vertex v E T. Two (k,n)-trees (T ,T , .v E V (T ) ) and (T1,'f[,,vE V(T ' ) ) are called isomorphic if there exists a system consisting of isomorphism of n-trees ~b • 7 --~ i rt and isomorphisms (o~. : T,, ---+ T~(~,) of ( k -

1,1nt(v))-trees. Let "fr(k,n) be the set o f isomorphism classes of (k, n )-trees. The sets Tr(k,n), k > 0 make up a cosimplicial set. The coface operators are generated by

operation o f substitution o f In( v )-labelled In(r)-stars at every vertex v o f a tree T.

The codegeneracy operators are induced by the composition o f trees o f the second

type. Let now ~ be an ~/-collection and let Q = (T, T~,,vE V(T) ) be a (k, n )-tree. I l k = 0

then we define ~(Q) by formula (2). By induction

g(Q) = @ &T~,). cCV(T)

Then we have a natural isomorphism

Vk(a)~ --~ I_I &r). TETr(k-l,n)

Moreover, it is evident that the morphisms of codegeneracy and coface in F*(~)~ are

induced by corresponding operators in Tr( , ,n) . It implies that the morphism

pk : D k ( F * ( g ) ) __+ F k - l ( g )

is an isomorphism on the summand. The proposition follows now from fibrantness of g:' and Lemma 2.3. []

Let K be a cocomplete monoidal locally fibrant strongly ,~/-tensored ~4-category,

be a cofibrant ,~/-operad and let U " C H A l g ~ --+ re(K) be a forgetful functor.

Definition 3.1. Let X be an object o f K, and let

~, v : ,~ ---+ E n d ( X )

be two g-algebra structures on X. We say that they are coherently isomorphic provided

there is an isomorphism r in CHAl9 ~ of (X,t/) and (X,v), such that U(r) is identity.

M.A. Batanin/Journal of Pure and Applied Alyebra 123 (1998) 67-103 87

L e m m a 3.1. Let (~, I~, e.) be a cqfibrant ~/-operad and let

~7, v • ~ ---+ E n d ( X )

be two .~-algebra structures on X, such that ~I is homotopic to v in Opera.. Then these structures are coherently isomorphic. This coherent isomorphism is not canonical and

depends on the choice o f homotopy class o f homotopies between q and v.

Convers'ely, iJ 'p and 7 7 are coherently isomorphic, then they are homotopic as

morphisms in Ope~..

Proof . Let X0 = (X, ~) and Xl -- (X, v). For every k _> 1 we have an i somorph i sm

A l g~( [ g]k Xo,XI ) ~ K_9,~([~]k- Ix0 ,Y 1 ).

Hence, to obtain a required i somorph i sm we have to construct some morph i sms

r k ' ( [ ~ ] ~ X ) x A k - - ~ x , k_> 1,

r ° = l x ,

sat isfying the coherency condit ions. W e shall do it by induction. Let us denote by 6n • A n -+ A ~ x A ~ the d iagonal mapping. The i terat ion o f 6n

generates a natural t ransformat ion

Dn : [4~]X x A" --~ [ ? ] ( X × A ~).

Note, that cofibrantness o f ~ and f ibrantness o f the col lec t ion E n d ( X ) imply the existence of a h o m o t o p y

r 1 " [ ? ] X x A 1 -+ X,

for which

r l ( l [ ~ B - x d °) = r/, (3)

r l ( l [ f d x x d l ) = v, (4)

r l ( z x 1~,) = l x × s o (5)

and the fo l lowing d iag ram commutes :

[ ~ f ] = X x A I I xa , A J A1 D, x l - , [g]ZX x x , [ g l ( [ a ' ] X x A t ) x zl I

[ ~ ' ] ( r I ) x I

[ 8 I X x A l rL ~' -- , X ~ [$ ' ]A" x A l

88 ~t/LA. Batanin/Journal of Pure and Applied Algebra 123 (1998) 6~103

Let now r k be defined for k <_ n - 1. Define r n by the following composition:

( [ ? ] " x ) x 3 " ~ ( [ # ] " X ) x J - ~ x ~t ~

rJ([#J(r ' ' - I x l ) ) [ , ~ ] ( [ o x ~ ] n - l x x A n - I ) x A 1 ' X .

D,,_ I x 1

It is not hard to verify, using formulas ( 3 ) - ( 5 ) and the diagram above, that the morphisms r k define a desired coherent morphism.

The converse statement is evident. []

Let now the tensor product in ~ commute with realization. Then the realization of a simplicial operad in ~ is again an operad. So the bar-construction B ( F , F , , ~ )

is an ,;/-operad, which we shall denote by F.~(d~). Note, that F ~ ( N ) is a cofibrant A~-operad if ~ is.

Example . The operad F ~ ( J { ) in ~9 ° admits the following description. Let n > 0. Then F ~ c ( J / ) n is the nerve of the partial ordered set of isomorphism classes of n-trees. The composition o i induces a multiplication by an obvious way.

The operad F.~(~ / [ ) contains some Aoc-suboperad ~ generated by reduced trees. More precisely, an n-tree T is reduced if there are at least two inputs at each vertex v C V ( T ) . We assume also that D and D ] are reduced, and that D is a single reduced 0-tree (compare with [14, 1.2.12]).

Let us give a description of ~ in terms of bracketing.

Let hn be the following category: (a) h0 contains only one object 1 and only identity morphism; (b) for n > 0 the objects of h~ are the sequences consisting of n formal symbols

al . . . an, several symbols 1 together with several pairs of brackets, such that the result of the corresponding formal product is meaningful (the repetition and the inverses of the generating elements a l , a 2 , . . , are forbidden).

For example h3 contains as the objects the following sequences:

ala2a3, l a l ( a 2 1 l a 3 ) , ( l ( a l a 2 ) l ) ( a 3 a l )

and so on. We have a naorphism from a sequence A0 to Aj if A0 may be obtained from A i by throwing off some pairs o f brackets and some symbols 1.

We have the functors

7 : hj , × . " × hi , × h~ ~ hj~+. . .+j , ,

which substitutes the object of hj, in place of the symbol ai in k~ (in brackets) and renumbers the symbols in the natural order. There is also a functor z : 1 ~ hi, which sends 1 to ai. We thus have a ~at-operad h.

We obtain an J - o p e r a d JC by applying the nerve functor to h. The geometric realization of ~ gives us some topological A~-operad H, which is a deformation

retract o f F ~ ( ~ # ) .

M.A. BataninlJournal of Pure and Applied Algebra 123 (1998) 67-103 89

The following result is a variant of a lifting theorem of [5].

Theorem 3.1. Let f : ~ ~ g' be a local weak equivalence oJ" cofibrant ~J-operads. I f in ,~/ the tensor product of weak equivalences is a weak equivalence, then Jor

every F~(g)-al~jebra structure ~I on X there exists an F~(g')-algebra structure v on X such that ( F ~ ( f ) * ( v ) , X ) is' coherently isomorphic to (q ,X) and any two such structures are coherently isomorphic.

Proof. Under the conditions of the theorem we obtain a homotopy equivalence

Oper_.~( k(~ ), End(X)) --+ Oper9.(Fk(g), End(X))

for every k _> 0. Hence, F ~ ( f ) induces a homotopy equivalence

Oper/~(F~(g ) ,End(X) ) -+ Oper>/,(F~(g),End(Y))

by Proposition 3.1. Thus the theorem follows from Lemma 3.1. []

Let g be a cofibrant A~-operad. By applying the theorem obtained to the aug-

mentation of ~ we see that one can replace every g-algebra structure on X by an

F~(.¢/)-algebra structure on X without altering the isomorphism class of X in the coherent homotopy category of A~:-monoids.

Thus we have

Corollary 3.1.1. The natural inclusion

CH Alg F~( //) C CH Mon

is an equivalence of categories over re(K).

4. Homotopy invariance

The material of this section is sufficiently known and goes back to [5]. The detailed proof of the theorem below in terms of strong homotopy algebras over monads was

given by Lada in [17]. We need, however, a slight generalization of his results, be-

cause we work with homotopy units. Furthermore, we do not use the notion of strong

homotopy algebra. It is not hard to verify that for an operad g the structure on X of a strong homotopy algebra (with homotopy unit) over monad [g] is exactly an F~(g)-algebra structure. Otherwise the methods of [17] work well and this allows us

to minimize the volume of the proof. Let o ~ be an s~-operad, then we have a morphism of operads

} ,~ : F~c ( ,d ) -~ g .

Let (X, q) be an g-algebra in some monoidal .~-category K and let f : X --~ Y be a

homotopy equivalence in K.

90 M.A. Batanin/Journal of Pure and Applied Algebra 123 (1998) 6~103

Theorem 4.1. I f the collections End(X) and End(Y) are fibrant and .~ is a cofibrant

opera& then f defines - a structure o f F~(E)-algebra on Y,

v : F ~ ( ~ ) --+ End(Y),

- an isomorphism,

( J ,1)) ~ (Y, v), ~,,*~((x, .

in CHAI9 F~(~), such that U(~b) = [ f ] , where U is the corresponding forgetful

fimctor.

To prove this theorem, we need some lemma about homotopy equivalences, which goes back to Vogt [28] and is considered in general form in [10, Proposition 2.3] (see also [17, Lemma 9.1]). Our present form of this lemma may be easily deduced from it.

Lemma 4.1. Let X, Y be two objects o f some simplicial category K, such that K__(X,X )

and K(K Y) are Kan simplicial sets. Let

f : X ~ Y, 9: Y - + X

be the inverse homotopy equivalences and let

~, c K_,(X,X)

be a homotopy, such that d 0 ( ~ l ) = lx, d l ( # l ) = g ' f . Then there exist a homotopy

~ ~ K_,(Y, Y)

and a 2-homotopy

~2 E K2(X,X) ,

such that

d o ( ~ l ) = f ' g , d l ( ~ i ) = 1y,

d0(~2) = ~1 " ~1,

d l (~2) = ~l,

d2(~2) = 9" ~l " f .

Proof of Theorem 4.1. To construct a desired Foo(g)-algebra structure on Y we show

that the date of the theorem generates some morphism of operads

h : F ~ ( E n d ( X ) ) -+ End(Y).

Then one can define v = h - F~(~/).


As was shown in Lemma 3.1 we have to construct a sequence of morphisms of ~J-collections,

h,, : Fn(End(X)) x A n ~ End(Y),

satisfying the coherence conditions• To do this we introduce some functor

F' " Coll ---+ Oper,

F(,~), = H ?(r). T~Tr(n), T~D I

So F ( 8 ) -~ F ' ( g ) [ I I .

Let us choose an inverse homotopy equivalence g " Y ~ X and a homotopy ¢1

between l x and g - f . By Lemma 4.1 we thus obtain a homotopy Ct and a 2-homotopy 32. We can consider the homotopies above as simplicial mappings

~l : Z~I ~ K j ( X , X ) , I//I ' AI ----+ K,9~,(Y, Y) , ~2 " A2 ~ K ~ , v ( X , X )

respectively. This allows us to construct h~ by induction• Let ho be the composition of the morphisms induced by f and g,

End(X) J~, -K(X, Y) g~, End(Y).

To define h~, n _> 1 we need some auxiliary morphisms

k, F ' (FnEnd(X)) z A 2 • ---+ F 'End(X) , n >_ O.

Let ko be the following composition:

F' (End(X) ) x A 2 lzprq~. F' (End(X) ) ~ End(X).

Let n > 1, k _> 0 and let T be a k-tree. Then the iterated diagonal A 2 --+ A 2 x A 2

induces the following morphism:

@ (F"End(X))I~,U,) x A 2 ~_

vE I'(T)

A 2 @ (F"End(X))l,,,u,) ,~ (FnEnd(X))ln,(o,,,(~)) x -+ (6) v E V(T), v~ Out( T )

@ (F"End(X) x A2)/,,,(u) @ (F"End(X))z~,{ou4r). vCV(T] v~Out(T)

Let v E V(T) ,v ¢ Out(T) then

(F"End(X))z~,(v) = H F"-I End(X)(T')" T' 6Tr([ln'(v)[)


Define some morphism on the summand with index T ~ ¢ D l by

I xs0(,fi ) ( F n - I E n d ( X ) ) ( T ') × A 2 ~ ( F n - l E n d ( X ) ) ( T ') × K, / . (X .X)

(Fn- l E n d ( X ) ) ( T ' ) , (7)

where the last morphism is induced by multiplication in E n d ( X ) (see the description of (FnEnd(X))in,(t,) in Proposition 3.1).

If T' = D 1, then define a morphism by the composition

~=~ K w(X ,X ) × ( F " - I E n d ( X ) ) ( T ') (Fn- l E n d ( X ) ) ( U ) × A 2 ~"

(Fn- t E n d ( X ) ) ( U ) . (8)

The morphisms (7),(8) define together the morphisms

(FnEnd(X))tn,l~,) x A 2 --~ (F~End(X))tn,(~,

which in composition with (6) give the desired morphism kn on the summand corresponding to a k-tree T.

Let now the morphisms hm be defined for m _< n - 1. Then define h, on the summand F' (Fn- I E n d ( X ) ) x A ~ by the following composition:

Ix(s,, I)'-I×s(J F/(F n - l E n d ( X ) ) × A n >

k,~_ I x l F~(Fn- IEnd(X) ) x A 2 × A n 1

F"- I E n d ( X ) x A n-l

On the summand I we define

l x ( s . I)~:-I I x A n > I x A l

]~;r-- I

: End(Y) .

ex~l K~..(Y, Y) × E n d l ( Y ) --~ End l (Y ) .

It is straightforward to verify, using the inductive description of F*(,~) given in Proposition 3.1, that the morphisms hn, n _> 0 satisfy the necessary coherent

conditions. For a construction of a coherent isomorphism q~ see [17, Theorem 6.1 (iii)]. To make

excuses to the reader, note that we do not use anywhere in our paper the second part

of the theorem. []

5. Cosimplicial d~-algebras and ~-coalgebras

Let A be a category with finite colimits and let X*, * be a bicosimplicial object in A. Then we can construct a cosimplicial object V ( X ) = V(X*,*) as follows [2]:

V(X)° = xO,O.


X7(X) n+l is the colimit of the diagram

x p , q +1

x P , q dl ~+'

> XP+I,q

T p + q = n .

See also [2] for the definition of cofaces and codegeneracies.

Remark. A construction o f this type was used for the first time in [1]. By this reason

2 7 is called in [12] the Artin-Mazur codiagonal functor. As is noted in [12], 27(X) is

none other than the left Kan extension along the ordinal sum functor. Let now K be a monoidal °~/-category with finite colimits. Then we can define the

tensor product ,~,~: of cosimplicial objects in K by the formula

X*®~x Y* = 27((X'~,x Y)*' * ),

where (X(~xY)* '* is a bicosimplicial object of the type

(X"~K Y)P'q = X P @ K Yq

with the obvious cofaces and codegeneracies [2].

Proposition 5.1. I f ~ x commutes with colimits, then the eategory cK o f cosimplic&l

objects o f K i6' a monoidal ~/-category with respect to the tensor product defined above and with the constant cosimplicial object I n = Ix as the unit object. Further- more, i l K is (strongly) ~/-tensored then cK is also (strongly) ~- tensored

Proof. Immediate from the definitions.

Let A n be the standard topological simplex

A " = ( ( u ] , . . . , u n ) c ~ ' l O < u ¿ <_... <Un < - l}.

Let A[s], 0 < s < vc be the cosimplicial space , consisting of the s-skeleton of A n

with usual cofaces and codegeneracies between them. We denote A[oc] by A.

Theorem 5.1. Let H be the operad defined in Section 3. Then Jbr each 0 < s < oc the cosimplicial space A[s] has an H-coalgebra structure in cJ~p. For all so <_ sl the natural inclusions A[s0] c A[sl] are the morphisrns of H-coal#ebras.

Proofl We shall prove this theorem for s = ec,, the other part of the theorem will

follow immediately.


To obtain a coaction o f H on A we have to construct the mappings

7m, n : A m × H n - ~ ( A @ . . . . @ A ) m , gtl, F/ > 0, Y

n

where we denote by ,~, the tensor product in c J o p . At the beginning we want to

describe some algorithm, which associates with every vertex A of Hn some subdivision o f the unit interval [0, 1].

Recall that a vertex A of Hn is a word consisting o f n symbols al . . . . ,a,,, l symbols

1 and some pairs of brackets. We call a subword of A a subsequence o f A, which

is either one of the symbols ai, 1 or is the expression taken in brackets in A. The set o f the subwords of A has a natural partial ordering. Namely, B0 > BI if BI is a

subword of B0. Let pl = 1 if A consists only of one symbol, otherwise let Pl be the number of the maximal elements of the poser o f proper (not equal to A ) subwords o f A. Consider the subdivision o f [0, 1] on Pl intervals [ ( j - 1)/'n,j/ 'n], j = 1, p l . We

associate the interval [(j - 1 ) / /n, j /n] with the j th maximal subword of A. We can

continue this process of subdivision by applying the same algorithm to the interval [( j - 1)/n, j / /n] and the set of proper subwords of A j for each j . Finally, we obtain a

subdivision of [0, 1] on n + l intervals, such that with every symbol ai, i = 1,n and every symbol 1 contained in A its own interval is associated. If the symbol has the

number j in A (in the natural order ) then we denote the corresponding interval by

[aj 1,ai], O < j _ < n + l , a o = O , q . + t = 1. For example, let A be a word (1 l ( a j a 2 ) l ) a 3 . Then the corresponding subdivision of

[0, 1] is

[0, 1/'8], [1,,/8, 1/"4], [1/4,5/16], [5/'16,3/8], [3/8, 1/2], [1,/2, 1].

A subdivision described allows us to define the following subdivision of A" on the subspaces prl,...,r,+l X "~n+l

, Z - a i = l r i = m :

P rl'''''r"+, = {u E Zlrt'lo" 0 ~ I.ll ~ " '" ~_ ldr I ~ 0"1 ~ Hrl+l

<_ "'" <_ U,',+r2 <_ a2 <_ "'" <_ U,, <_ 1}.

Let now [aj~_l,aj~] be an interval associated with symbol ak, k = l,n. Define a

map

)'re, n : Am ---+ (/ t @ . . . . @: h )m

n

on the vertex A by the formula

7m, n (u) = b('~l(u) . . . . . 9:n(u)), u E P .......... ,+,,

where b is the canonical morphism in colimit and

2k : pr~,...,r.+t ~ ArJk

M.A. Batanin/Journal of Pure and Applied Alyebra 123 (1998) 67-103 95

is defined by

7gk(bl) = ( blr,+'"+ri~-,+l ~ lYJ~ -1 blr,+'"+ri,. -- GJk - I ) , . . . ,

\ aj~ - crj~ 1 crj~ - a j , I

By way of illustration we present )'2,3, corresponding to the vertex A from the above example, in the following picture:

al a2

Let now

A0 ~-- " '" +-Ak

a 3

a 3

a2 a l

/ /

/

be a k-simplex of H~. Then we can construct the k + 1 subdivisions of [0, 1 ] on n + l intervals by applying the process above to each Ai and assuming that we insert a trivial interval [0, or] in place of the symbols I thrown off. Let us demonstrate this on

an example. Consider the following 2-simplex o f / / 3 :

( l l ( a l a 2 ) l ) a 3 ~-- ( ( a l a 2 ) l ) a 3 ~ - a l a 2 a 3 .

Then we obtain 3 subdivisions of [0, 1] on 6 intervals:

[0, 1/8], [1/"8, 1,,'4], [1/'4,5/16], [5/16,3//8], [3,,'8, 1/'2], [1,/'2, 1].

[0,0], [0,0], [0,1/8], [1//8,1//4], [1/'4,1/'2], [1/'2,11.

[0,0], [0,0], [0,1/3], [1/3,2/3], [2/3,2/3], [2/3,1].

Return to the general construction. Let

[(Tj_I(S),~rj(S)] , 1 < _ j < _ n + l , 0 < s < k

be the interval corresponding to a symbol with number j in the sth subdivision. Let

t = (to . . . . . tk) E A k and let

O i ( t ) = ti+l - ti, t o = O , t~+l = 1, O < i < k.

96 M.A. Batanin/Journal of Pure and Applied Aloebra 123 (1998) 67-103

Then define

p~ ............ ( t ) = A ~ 0 (u , t ) C A m x <_ ul <_ " " <_ u~, <_ O,(t)~rl(s) s = 0

k

< Ur,+l _< "'" < Ur,+~_~ < ~ O , ( t ) a 2 ( S ) <_ " " < U,,, < 1

x=0

Then we put ;:m,,; on the simplex A0 +- • • • ~ Ak of H~ equal to

7m. , , (u , t )=b(x l (u , t ) . . . . . ~"(u,t)), (u , t ) E P ............ ( t ) ,

.

where ~i(u,t) is defined by the same formulas as cd(u) but instead o f ~j, we use k

~s=00s ( t )~ j , ( s ) . It is straightforward to check that we thus obtain a coaction o f H on A. []

Remark. Our subdivision U ' .......... ( t ) is an immediate generalization o f subdivision of A" introduced in [18, 8, 9, 2]. For example, the comultiplication

p : A - - - + A @ A

is the image under 7 of the vertex aja2 o f / / 2 [2].

Corollary 5.1.1. Consider 3-op as an Y-category. Then the cosimplicial topological space A[s], 0 < s < cx~ has an Yi~-coalgebra structure.

Let now K be a complete monoidal J - ca tegory . Then for any cosimplicial objects M* of K and E* of ~ we define the realization of M* with respect to E* by

(M*) e* = f p ( M P ) E;'.

Theorem 5.2. Let o~1 and ~2 be two s~/-operads. I f ®x commutes with finite colimits then for any gl-algebra M* in cK and ~2-coalgebra E* in c~4 the realization, (M*) E*, is' and (o~1 ~ : j g2)-algebra in K.

Furthermore, this construction is functorial, that is we have the realization as an

.yJ-functor

(Coalg~2(c.~£)) ~.~/(Alg~'(cK)) ~ Alg a' ®'Jz'-~(K).

Proof. Let us show firstly that there exists an ,~/-natural transformation

Z" (N* ) E ~ K ( M * ) F* --* (N* @cKM* ) E*'~rjF*,

which makes monoidal the functor of realization.

M.A. Batanin/Journal of Pure and Applied Algebra 123 (1998) 6~I03

Indeed, we have an obvious transformation

"~o " (N*)E*@K(M*) F* = fp(NP)EP@K fq(Mq) F'

--~ ~,,q (NP)E"GK(Mq)F" ---~ fp, q (NPc~K,~4q) E''®:jF'.

Let now X be an object of K, then

K (X,~,q(NP@KMq) Ej'~jF('} ~ L , q K( X, NP'~)KMq) U'~F'')

fp ~g~,(Ep,~.~¢Fq,K(X, NP@KMq)) v ~ . . ~_ c~4(V((E,~ ,~F) ' ), ,q

~7(K(X, (N+KM)*' * ))) ~ c~/(~7((E ~,~F)*' * ), K__(X, V((N+KM)*' * )))

= c~/(E* ®~/F* , K(X, N*,~cKM* ))

97

F = Jn ~ ( ( E * ~,~.~ F* )", K(X, (N*,N,,.KM*)~ )) "~ K(X, (N* @cxM* )E* ~,,¢F* ).

Thus by the enriched Yoneda lemma [16] we have a morphism

" f (NP@'KMq)E~'Y)~F'I ~ (N*@cKM*)E*®"~F~" Jp ,q

Then we put ~ = rt • z0. It is not hard to check that r has the desired properties. Let m be the following ~4-natural transformation:

cK(m*, N* ) ~.~l c~(E* , F* ) ~ K((M* )F* (N*)F* ) ~.~ K_((N* )F*, (N*)E*)

_L+ K__((M* )F*, (N*)E*),

where p is the composition in K. Using the transformations m and r we obtain

cK((M* )g, N* ) ®.~¢ c ~ ( E * , (F*)J) m K(((M* )])~F* )', (N*)E')

) K(((M*) F* )J,(N*)E*).

This provides us with a natural transformation

r : c-~(M*, N* ) c~,~ c ~ ° ( E *, F* ) ~ ~( (M* )g*, (N*)E* ).

Finally, using the ~l-algebra structure on M* and the ~5-coalgebra structure on E*, we obtain a morphism of ~]-collections

~1 ~.~]~2 ~ End(M*)'~.~/ End°(E*) ~ ~ End((M*)E*) .

It is straightforward to verify that this morphism is a morphism of operads. The naturality of the constructions involved gives us the second part of the theorem. []

98 A'L A. Batanin/Journal o f Pure and Applied Algebra 123 (1998) 67-103

Corollary 5.2.1. Let s~" be a structure of a model Quillen category. I f in ~/ the tensor product of weak equivalences is a weak equivalence, then under the conditions oJ" Theorem 5.2 the realization of an A~-monoM in cK with respect to an A ~ - comonoM in c~J is an A~-monoid in K.

6. Some applications to coherent homotopy theory

The results obtained permit us to generalize the construction o f the coherent homo-

topy category o f a comonad (a monad) introduced in [2]. Let K be an ,~-category and let (L,p,c) be an ..~-comonad on K. Let E be an ,~/-operad and let N be an g-coalgebra in c~4. Then we can consider an ,~/-endodistributor on K,

K(L, (X) , y)N -_ ( (~L, )N(X ' y)

(see Section 2 for the definition of a simplicial ~z/-functor L , ) .

Proposition 6.1. The distributor

(¢~)x

has a natural structure of an g-algebra in Dist(K,K). If, in addition, K is cocomplete, then the endofimctor

f L, ~,~jN(X) = LP+I(X)~=./N p

has a natural d-coalgebra structure in F(K, K).

Proof. We can apply Theorem 5.2 if we prove that the cosimplicial ~-distr ibutor ~b L*

has a structure of a monoid in Dist(K,K). Define a multiplication as follows:

F~ pq : K_(L p+ I (X), Y) ~,~ K(L q+l (Y), Z)

K(Lq+ILp+I(X),Lq+I(y))@,~/ K ( L q - I ( y ) , z ) ---+

L~I .p.L t K ( L q + I L p + I ( X ) , Z ) - I K ( L q + P + I ( X ) , Z ) .

The counit e, generates obviously some cosimplicial morphism 1 --~ q5 L*. It is easy to check that we thus obtain a needed monoid structure (see also [2] for the dual case of

a monad). The second part o f the proposition follows from Proposition 1.4. []

To relate this construction with that o f coherent homotopy category we give the

following

Definition 6.1. We call the homotopy category o f an A s - g r a p h G in ~ the category

with objects those o f G and with n0.~,(/,~/, G(X, Y)) as the morphisms from X to Y.


The composition and identity morphisms are induced by the Ao~-monoid structure in the obvious way.

The proposition just established and Theorem 5.1 give

Theorem 6.1. Let (L, p, e) be a .~bp-comonad on a :Y-op-category K. For each 0 < s < ,~ the distributor (OL*) 4q has a natural structure of H-algebra in Dist(K,K). A natural morphism of distributors

I -~ (q~L.)AN,

generated by ~, is the morphism of H-algebras. I f K is cocomplete, then the endoJimctor L, x A[s] has a natural H-coalgebra

structure in F(K,K). The coherent homotopy category of (L,p,e) with respect to A[s] is the homotopy

categoo, of A ~-graph ((dp L* )d)~[sl.

Let now K be a cocomplete ~Y-bp-category and let (R,p,~I) be a J'~p-monad on K. Let, in addition, R commute with the functor of geometric realization. Then we have

Corollary 6.1.1. May's bar-construction [20, 21]

B(R,R,X) = J l Bp(R,R,X) × A p

is the H-coal,qebra in the category of Jbp-endofunctor on the category of R-algebras. The coherent homotopy category of R-algebras is the homotopy cateqory of an

A~-graph, which as,s'igns to R-algebras X and Y a topological space

AlgR(B,(R,R,X), Y).

The simplicial analogue of the above theory is of special interest. But we cannot develop it immediately, because the cosimplicial simplicial set A has no A~-comonoid structure. This is the same kind of difficulty which was considered in [2]. To overcome it we use the construction from example 4, Section 1.

Let C be a simplicial category, then one can consider the category of topological distributors Dist~-op(C, C) considering .~p as a simplicial category. The fibrewise singular complex functor provides us some monoidal (not strong) ~-functor

S : Dist¢,,p(C, C) ~ Dist(C, C).

Proposition 6.2. Let (T*,/t*,~l*) be a monoid in cDistj-op(C,C). Then a simplicial endodistributor Tot( S( T* ) ) has a natural structure of Jr<algebra in Dist( C, C ). More- over, a morphism

q~ :I ~ Tot(S(T*))

induced by unit 0" is a morphism of .2,u{'-algebras.

100 M.A. BatanNIJournal of Pure and Applied Algebra 123 (1998) 6~103

Proof. Note that the singular complex functor maps the H-algebras in Dist~-op(C,C) to the ,.~-algebras in D&t(C, C). This follows from the monoidality of S and the tact that the natural morphism J4 ~ --+ S(H) is a morphism of ,9~-operads.

Let X, Y be objects of C, then

Tot(S(T*))(X, Y) = f , ,~(A(n), S(Tn(X, Y)))

f ~ "A" ~"X. ~- .yop~,A ,1 t , Y ) ) = S ( ( T * ) ~ ( X , Y ) ) .

But (T*) A is an H-algebra and hence Tot(S(T*)) ~_ S((T*) ~) is an fft~-algebra. []

Remark. The proposition just proved is very useful in categorical strong shape theory developed in [3].

We are now ready to prove a simplicial analogue of Theorem 6.1.

Theorem 6.2. Let (L, p, g) be an .Y-comonad on an ,~-category K. I f

K__(L,(X), Y)

is a fibrant cosimplicial simplicial set for ever), X, Y E ob(K) then the locally Kan graph Tot(O c*)a has a natural structure of H-algebra and the coherent homotopy category CHLoe - K is isomorphic to the homotopy category of this graph.

Proof. From the conditions of the theorem we have a homotopy equivalence of Kan simplicial sets

Tot(~ L~ )(X, Y) ~ Tot(S(IOL*(~ Y)]))

where ] - I is fibrewise (with respect to cosimplicial structure) geometric realization. The topological cosimplicial distributor ]q5 c* ] is a cosimplicial monoid. Then by Proposition 6.2 Tot(S( IqSL" 1)) has a structure of ~-algebra. Now by Theorems 3.1 and 4.1 the locally Kan graph Tot(O L* )a has a structure of an F~G~/)-algebra and hence of ~'~-algebra. 7]

Corollary 6.2.1. Let K be a locally Kan monoidal J-category with strong ,9~-tensori - zation and let ~ be an Y-operad.

Then the coherent homotopy category of ~-algebras is a homotopy category of the locally Kan A~-graph Tot(O I¢]* )d.

Finally, we can apply Theorem 6.2 to show that the coherent homotopy category of the diagrams may be obtained by a natural way as a homotopy category of some

locally Kan Ao~-graph. Recall now some definitions of coherent homotopy theory developed in [2, 8-10, 12].

M.A. Batanin/Journal of Pure and Applied A19ebra 123 (1998) 67 103 101

The following construction is extracted from [12]. Let A be a small 5P-category. For

objects X, Y of A form the bisimplicial set 7~(X, Y) defined by

~(X, Y)~,, = H A(X, Ao) x A_(Ao,A.) × . x A(A,,, Y) AII,...,A.

where

di : ~P(X, Y)~,, ~ ~(X, Y)n-I,*

is defined by composition in A,

d ( A i - l , A i ) × A(Ai, Ai+I ) ~ d ( A i - l , A i + l ),

and si : 7J(X, Y)n,* ---+ 7~(X, Y),,+I,* by the canonical morphism

A ° -~ A(Ai,Ai).

Set now.d(X, Y) = Diag(~(X, Y)).

Definition 6.2. Let B be a complete Y-category, with cotensorization

( - ) ( - ) : B × 5 ,'°p ---+ B,

and let

T :A°P x A - + B

be an 5'~-functor.

The simplicially coherent end of T will be the object fA T(A,A) of B defined by

fA T(A,A)= fA~.,,xA T(A',A")~i(A'"4").

As was established in [12], for the coherent end the cosimplicial replacement formula and a universal property like that of homotopy limit are fulfilled.

Let A be a small simplicial category, and let F, G : A --, K be two simplicial functors to an Y-category K. Then we can consider as the simplicial set of coherent natural

transformations from F to G the coherent end [8, 10, 12]:

Coh(F, G) = ~ K(F(2), G(2)).

As was shown in [12], for weakly locally Kan category K one can define a composition of coherent transformations, which is associative up to coherent homotopy. On the level

of the set of homotopy classes of coherent transformations we thus get the category

coh(A,K) called the coherent homotopy category of the diagrams of the type A.

Theorem 6.3. I f K is cocomplete locally Kan ~-category then the assignment

F, G H Coh(F, G)


has the structure of a locally Kan simplicial A.~-graph. The coherent homotopy category of the diagrams coh(A,K) is the homotopy category of this graph.

Proof . Consider the discretization Ad of the ca tegory A and the inclusion i : Aa --+ A. Denote F(A , K ) the ,~ -ca tegory o f Y- func to r s f rom A to K. Then i gives rise to a

pair o f Y - a d j o i n t functors [16]

i* : F ( A , K ) --~ F(Ad,K), Lani : F(Ad,K) ~ F(A,K) ,

and thus we have some ,Y-comonad (L,p,~O on F(A,K) . The p roof consists now in verif icat ion that

Coh(F, G) ~_ Tot(~ L* )(F, G),

which is an immedia te corol lary o f the cos impl ic ia l replacement formula [12]. Further-

more, as K is local ly Kan the cosimplic ia l s implicial set OL*(F,G) is fibrant [12, 2]

and so we are in the condi t ions o f Theorem 6.2. []

Acknowledgements

I would like to thank T. Porter for his encouragement and attention to my work and

J.-M. Cordier for the st imulating discussions during m y visit to Amiens in the spring

o f 1994.

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Homotopy coherent category theory and A -structures in ...dodo.pdmi.ras.ru/~topology/books/batanin.pdf · categorical algebra, whereas the alternative ones use the methods of homotopy